Ok, but I'm studying classical mechanics so I think I have to use another approach...

I'm not aware of another approach. You start off with a certain amount of energy, which is a constant in a conserved system. But the classical turning points are when [tex]\dot{x} = 0[\tex], and then a moment later the velocity changes signs (i.e. the particle goes from going to the right to going to the left), so then you do what Dexter suggests, and hopefully understand why you're doing it.